3frac (problem 3.3.3)

Percentage Accurate: 84.5% → 99.2%

Time: 11.1s

Alternatives: 7

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))

C

double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 / (x + 1.0d0)) - (2.0d0 / x)) + (1.0d0 / (x - 1.0d0))
end function

Java

public static double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Python

def code(x):
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0))

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))

C

double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 / (x + 1.0d0)) - (2.0d0 / x)) + (1.0d0 / (x - 1.0d0))
end function

Java

public static double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Python

def code(x):
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0))

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ t_1 := x \cdot x - x\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-10} \lor \neg \left(t_0 \leq 10^{-31}\right):\\ \;\;\;\;\frac{t_1 + \left(1 + x\right) \cdot \left(x - \mathsf{fma}\left(x, 2, -2\right)\right)}{\left(1 + x\right) \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (/ 1.0 (+ 1.0 x)) (/ 2.0 x)) (/ 1.0 (+ x -1.0))))
        (t_1 (- (* x x) x)))
   (if (or (<= t_0 -2e-10) (not (<= t_0 1e-31)))
     (/ (+ t_1 (* (+ 1.0 x) (- x (fma x 2.0 -2.0)))) (* (+ 1.0 x) t_1))
     (/ 2.0 (pow x 3.0)))))

C

double code(double x) {
	double t_0 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	double t_1 = (x * x) - x;
	double tmp;
	if ((t_0 <= -2e-10) || !(t_0 <= 1e-31)) {
		tmp = (t_1 + ((1.0 + x) * (x - fma(x, 2.0, -2.0)))) / ((1.0 + x) * t_1);
	} else {
		tmp = 2.0 / pow(x, 3.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(Float64(1.0 / Float64(1.0 + x)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	t_1 = Float64(Float64(x * x) - x)
	tmp = 0.0
	if ((t_0 <= -2e-10) || !(t_0 <= 1e-31))
		tmp = Float64(Float64(t_1 + Float64(Float64(1.0 + x) * Float64(x - fma(x, 2.0, -2.0)))) / Float64(Float64(1.0 + x) * t_1));
	else
		tmp = Float64(2.0 / (x ^ 3.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] - x), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e-10], N[Not[LessEqual[t$95$0, 1e-31]], $MachinePrecision]], N[(N[(t$95$1 + N[(N[(1.0 + x), $MachinePrecision] * N[(x - N[(x * 2.0 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -2.00000000000000007e-10 or 1e-31 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 99.6%

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]
   4. Step-by-step derivation

      1. frac-sub 99.6%

         \[\leadsto \frac{1}{1 + x} - \color{blue}{\frac{2 \cdot \left(x + -1\right) - x \cdot 1}{x \cdot \left(x + -1\right)}} \]

      2. frac-sub 100.0%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(x \cdot \left(x + -1\right)\right) - \left(1 + x\right) \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)}} \]

      3. *-un-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{x \cdot \left(x + -1\right)} - \left(1 + x\right) \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]

      4. distribute-rgt-in 100.0%

         \[\leadsto \frac{\color{blue}{\left(x \cdot x + -1 \cdot x\right)} - \left(1 + x\right) \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]

      5. neg-mul-1 100.0%

         \[\leadsto \frac{\left(x \cdot x + \color{blue}{\left(-x\right)}\right) - \left(1 + x\right) \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]

      6. sub-neg 100.0%

         \[\leadsto \frac{\color{blue}{\left(x \cdot x - x\right)} - \left(1 + x\right) \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]

      7. *-rgt-identity 100.0%

         \[\leadsto \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(2 \cdot \left(x + -1\right) - \color{blue}{x}\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]

      8. distribute-rgt-in 100.0%

         \[\leadsto \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\color{blue}{\left(x \cdot 2 + -1 \cdot 2\right)} - x\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]

      9. metadata-eval 100.0%

         \[\leadsto \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\left(x \cdot 2 + \color{blue}{-2}\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]

      10. metadata-eval 100.0%

          \[\leadsto \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\left(x \cdot 2 + \color{blue}{\left(-2\right)}\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]

      11. fma-def 100.0%

          \[\leadsto \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\color{blue}{\mathsf{fma}\left(x, 2, -2\right)} - x\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]

      12. metadata-eval 100.0%

          \[\leadsto \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\mathsf{fma}\left(x, 2, \color{blue}{-2}\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]

      13. distribute-rgt-in 100.0%

          \[\leadsto \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \color{blue}{\left(x \cdot x + -1 \cdot x\right)}} \]

      14. neg-mul-1 100.0%

          \[\leadsto \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x + \color{blue}{\left(-x\right)}\right)} \]

      15. sub-neg 100.0%

          \[\leadsto \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \color{blue}{\left(x \cdot x - x\right)}} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \frac{\left(x \cdot x - x\right) - \color{blue}{\left(x + 1\right)} \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

      2. remove-double-neg 100.0%

         \[\leadsto \frac{\left(x \cdot x - x\right) - \left(\color{blue}{\left(-\left(-x\right)\right)} + 1\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

      3. metadata-eval 100.0%

         \[\leadsto \frac{\left(x \cdot x - x\right) - \left(\left(-\left(-x\right)\right) + \color{blue}{\left(--1\right)}\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

      4. distribute-neg-in 100.0%

         \[\leadsto \frac{\left(x \cdot x - x\right) - \color{blue}{\left(-\left(\left(-x\right) + -1\right)\right)} \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

      5. neg-mul-1 100.0%

         \[\leadsto \frac{\left(x \cdot x - x\right) - \left(-\left(\color{blue}{-1 \cdot x} + -1\right)\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

      6. *-commutative 100.0%

         \[\leadsto \frac{\left(x \cdot x - x\right) - \left(-\left(\color{blue}{x \cdot -1} + -1\right)\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

      7. fma-udef 100.0%

         \[\leadsto \frac{\left(x \cdot x - x\right) - \left(-\color{blue}{\mathsf{fma}\left(x, -1, -1\right)}\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

      8. distribute-lft-neg-in 100.0%

         \[\leadsto \frac{\left(x \cdot x - x\right) - \color{blue}{\left(-\mathsf{fma}\left(x, -1, -1\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)\right)}}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

      9. distribute-lft-neg-in 100.0%

         \[\leadsto \frac{\left(x \cdot x - x\right) - \color{blue}{\left(-\mathsf{fma}\left(x, -1, -1\right)\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

      10. fma-udef 100.0%

          \[\leadsto \frac{\left(x \cdot x - x\right) - \left(-\color{blue}{\left(x \cdot -1 + -1\right)}\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

      11. *-commutative 100.0%

          \[\leadsto \frac{\left(x \cdot x - x\right) - \left(-\left(\color{blue}{-1 \cdot x} + -1\right)\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

      12. neg-mul-1 100.0%

          \[\leadsto \frac{\left(x \cdot x - x\right) - \left(-\left(\color{blue}{\left(-x\right)} + -1\right)\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

      13. distribute-neg-in 100.0%

          \[\leadsto \frac{\left(x \cdot x - x\right) - \color{blue}{\left(\left(-\left(-x\right)\right) + \left(--1\right)\right)} \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

      14. remove-double-neg 100.0%

          \[\leadsto \frac{\left(x \cdot x - x\right) - \left(\color{blue}{x} + \left(--1\right)\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

      15. metadata-eval 100.0%

          \[\leadsto \frac{\left(x \cdot x - x\right) - \left(x + \color{blue}{1}\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

      16. +-commutative 100.0%

          \[\leadsto \frac{\left(x \cdot x - x\right) - \left(x + 1\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\color{blue}{\left(x + 1\right)} \cdot \left(x \cdot x - x\right)} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\left(x \cdot x - x\right) - \left(x + 1\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(x + 1\right) \cdot \left(x \cdot x - x\right)}} \]

   if -2.00000000000000007e-10 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 1e-31

   1. Initial program 66.8%

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

   3. Simplified 66.8%

      \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]

   4. Taylor expanded in x around inf 99.0%

      \[\leadsto \color{blue}{\frac{2}{{x}^{3}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -2 \cdot 10^{-10} \lor \neg \left(\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 10^{-31}\right):\\ \;\;\;\;\frac{\left(x \cdot x - x\right) + \left(1 + x\right) \cdot \left(x - \mathsf{fma}\left(x, 2, -2\right)\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \end{array} \]

Alternative 2: 99.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ t_1 := 1 - x \cdot x\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 10^{-31}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_1} - \left(\left(\frac{2}{x} + \frac{-1}{x + -1}\right) + \frac{x}{t_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (/ 1.0 (+ 1.0 x)) (/ 2.0 x)) (/ 1.0 (+ x -1.0))))
        (t_1 (- 1.0 (* x x))))
   (if (<= t_0 -2e-10)
     t_0
     (if (<= t_0 1e-31)
       (/ 2.0 (pow x 3.0))
       (- (/ 1.0 t_1) (+ (+ (/ 2.0 x) (/ -1.0 (+ x -1.0))) (/ x t_1)))))))

C

double code(double x) {
	double t_0 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	double t_1 = 1.0 - (x * x);
	double tmp;
	if (t_0 <= -2e-10) {
		tmp = t_0;
	} else if (t_0 <= 1e-31) {
		tmp = 2.0 / pow(x, 3.0);
	} else {
		tmp = (1.0 / t_1) - (((2.0 / x) + (-1.0 / (x + -1.0))) + (x / t_1));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((1.0d0 / (1.0d0 + x)) - (2.0d0 / x)) + (1.0d0 / (x + (-1.0d0)))
    t_1 = 1.0d0 - (x * x)
    if (t_0 <= (-2d-10)) then
        tmp = t_0
    else if (t_0 <= 1d-31) then
        tmp = 2.0d0 / (x ** 3.0d0)
    else
        tmp = (1.0d0 / t_1) - (((2.0d0 / x) + ((-1.0d0) / (x + (-1.0d0)))) + (x / t_1))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	double t_1 = 1.0 - (x * x);
	double tmp;
	if (t_0 <= -2e-10) {
		tmp = t_0;
	} else if (t_0 <= 1e-31) {
		tmp = 2.0 / Math.pow(x, 3.0);
	} else {
		tmp = (1.0 / t_1) - (((2.0 / x) + (-1.0 / (x + -1.0))) + (x / t_1));
	}
	return tmp;
}

Python

def code(x):
	t_0 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0))
	t_1 = 1.0 - (x * x)
	tmp = 0
	if t_0 <= -2e-10:
		tmp = t_0
	elif t_0 <= 1e-31:
		tmp = 2.0 / math.pow(x, 3.0)
	else:
		tmp = (1.0 / t_1) - (((2.0 / x) + (-1.0 / (x + -1.0))) + (x / t_1))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(Float64(1.0 / Float64(1.0 + x)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	t_1 = Float64(1.0 - Float64(x * x))
	tmp = 0.0
	if (t_0 <= -2e-10)
		tmp = t_0;
	elseif (t_0 <= 1e-31)
		tmp = Float64(2.0 / (x ^ 3.0));
	else
		tmp = Float64(Float64(1.0 / t_1) - Float64(Float64(Float64(2.0 / x) + Float64(-1.0 / Float64(x + -1.0))) + Float64(x / t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	t_1 = 1.0 - (x * x);
	tmp = 0.0;
	if (t_0 <= -2e-10)
		tmp = t_0;
	elseif (t_0 <= 1e-31)
		tmp = 2.0 / (x ^ 3.0);
	else
		tmp = (1.0 / t_1) - (((2.0 / x) + (-1.0 / (x + -1.0))) + (x / t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-10], t$95$0, If[LessEqual[t$95$0, 1e-31], N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$1), $MachinePrecision] - N[(N[(N[(2.0 / x), $MachinePrecision] + N[(-1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -2.00000000000000007e-10

   1. Initial program 99.6%

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

   if -2.00000000000000007e-10 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 1e-31

   1. Initial program 66.8%

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

   3. Simplified 66.8%

      \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]

   4. Taylor expanded in x around inf 99.0%

      \[\leadsto \color{blue}{\frac{2}{{x}^{3}}} \]

   if 1e-31 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 99.6%

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]
   4. Step-by-step derivation

      1. flip-+ 99.6%

         \[\leadsto \frac{1}{\color{blue}{\frac{1 \cdot 1 - x \cdot x}{1 - x}}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

      2. sub-neg 99.6%

         \[\leadsto \frac{1}{\frac{1 \cdot 1 - x \cdot x}{\color{blue}{1 + \left(-x\right)}}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

      3. metadata-eval 99.6%

         \[\leadsto \frac{1}{\frac{1 \cdot 1 - x \cdot x}{\color{blue}{\left(--1\right)} + \left(-x\right)}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

      4. distribute-neg-in 99.6%

         \[\leadsto \frac{1}{\frac{1 \cdot 1 - x \cdot x}{\color{blue}{-\left(-1 + x\right)}}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

      5. +-commutative 99.6%

         \[\leadsto \frac{1}{\frac{1 \cdot 1 - x \cdot x}{-\color{blue}{\left(x + -1\right)}}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

      6. associate-/r/ 99.6%

         \[\leadsto \color{blue}{\frac{1}{1 \cdot 1 - x \cdot x} \cdot \left(-\left(x + -1\right)\right)} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

      7. metadata-eval 99.6%

         \[\leadsto \frac{1}{\color{blue}{1} - x \cdot x} \cdot \left(-\left(x + -1\right)\right) - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

      8. +-commutative 99.6%

         \[\leadsto \frac{1}{1 - x \cdot x} \cdot \left(-\color{blue}{\left(-1 + x\right)}\right) - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

      9. distribute-neg-in 99.6%

         \[\leadsto \frac{1}{1 - x \cdot x} \cdot \color{blue}{\left(\left(--1\right) + \left(-x\right)\right)} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

      10. metadata-eval 99.6%

          \[\leadsto \frac{1}{1 - x \cdot x} \cdot \left(\color{blue}{1} + \left(-x\right)\right) - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

      11. sub-neg 99.6%

          \[\leadsto \frac{1}{1 - x \cdot x} \cdot \color{blue}{\left(1 - x\right)} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

   5. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{1}{1 - x \cdot x} \cdot \left(1 - x\right)} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
   6. Step-by-step derivation

      1. associate-*l/ 99.6%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(1 - x\right)}{1 - x \cdot x}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

      2. *-lft-identity 99.6%

         \[\leadsto \frac{\color{blue}{1 - x}}{1 - x \cdot x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{1 - x}{1 - x \cdot x}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
   8. Step-by-step derivation

      1. *-un-lft-identity 99.6%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(1 - x\right)}}{1 - x \cdot x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

      2. associate-*l/ 99.6%

         \[\leadsto \color{blue}{\frac{1}{1 - x \cdot x} \cdot \left(1 - x\right)} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

      3. sub-neg 99.6%

         \[\leadsto \frac{1}{1 - x \cdot x} \cdot \color{blue}{\left(1 + \left(-x\right)\right)} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

      4. distribute-rgt-in 99.6%

         \[\leadsto \color{blue}{\left(1 \cdot \frac{1}{1 - x \cdot x} + \left(-x\right) \cdot \frac{1}{1 - x \cdot x}\right)} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

      5. *-un-lft-identity 99.6%

         \[\leadsto \left(\color{blue}{\frac{1}{1 - x \cdot x}} + \left(-x\right) \cdot \frac{1}{1 - x \cdot x}\right) - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

   9. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\left(\frac{1}{1 - x \cdot x} + \left(-x\right) \cdot \frac{1}{1 - x \cdot x}\right)} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
   10. Step-by-step derivation

       1. associate--l+ 99.7%

          \[\leadsto \color{blue}{\frac{1}{1 - x \cdot x} + \left(\left(-x\right) \cdot \frac{1}{1 - x \cdot x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)\right)} \]

       2. un-div-inv 99.7%

          \[\leadsto \frac{1}{1 - x \cdot x} + \left(\color{blue}{\frac{-x}{1 - x \cdot x}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)\right) \]

   11. Applied egg-rr 99.7%

       \[\leadsto \color{blue}{\frac{1}{1 - x \cdot x} + \left(\frac{-x}{1 - x \cdot x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -2 \cdot 10^{-10}:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 10^{-31}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - x \cdot x} - \left(\left(\frac{2}{x} + \frac{-1}{x + -1}\right) + \frac{x}{1 - x \cdot x}\right)\\ \end{array} \]

Alternative 3: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ 1.0 x)) (/ 2.0 x)) (/ 1.0 (+ x -1.0))))

C

double code(double x) {
	return ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 / (1.0d0 + x)) - (2.0d0 / x)) + (1.0d0 / (x + (-1.0d0)))
end function

Java

public static double code(double x) {
	return ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
}

Python

def code(x):
	return ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0))

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / Float64(1.0 + x)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.6%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

2. Final simplification 84.6%

   \[\leadsto \left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \]

Alternative 4: 75.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.15 \cdot 10^{+77}\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.15e+77)))
   (/ -1.0 (* x x))
   (- 1.0 (/ 2.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.15e+77)) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = 1.0 - (2.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.15d+77))) then
        tmp = (-1.0d0) / (x * x)
    else
        tmp = 1.0d0 - (2.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.15e+77)) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = 1.0 - (2.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.15e+77):
		tmp = -1.0 / (x * x)
	else:
		tmp = 1.0 - (2.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.15e+77))
		tmp = Float64(-1.0 / Float64(x * x));
	else
		tmp = Float64(1.0 - Float64(2.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.15e+77)))
		tmp = -1.0 / (x * x);
	else
		tmp = 1.0 - (2.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.15e+77]], $MachinePrecision]], N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.14999999999999997e77 < x

   1. Initial program 78.9%

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

   3. Simplified 78.9%

      \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]
   4. Step-by-step derivation

      1. flip-+ 21.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{1 \cdot 1 - x \cdot x}{1 - x}}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

      2. sub-neg 21.5%

         \[\leadsto \frac{1}{\frac{1 \cdot 1 - x \cdot x}{\color{blue}{1 + \left(-x\right)}}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

      3. metadata-eval 21.5%

         \[\leadsto \frac{1}{\frac{1 \cdot 1 - x \cdot x}{\color{blue}{\left(--1\right)} + \left(-x\right)}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

      4. distribute-neg-in 21.5%

         \[\leadsto \frac{1}{\frac{1 \cdot 1 - x \cdot x}{\color{blue}{-\left(-1 + x\right)}}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

      5. +-commutative 21.5%

         \[\leadsto \frac{1}{\frac{1 \cdot 1 - x \cdot x}{-\color{blue}{\left(x + -1\right)}}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

      6. associate-/r/ 19.5%

         \[\leadsto \color{blue}{\frac{1}{1 \cdot 1 - x \cdot x} \cdot \left(-\left(x + -1\right)\right)} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

      7. metadata-eval 19.5%

         \[\leadsto \frac{1}{\color{blue}{1} - x \cdot x} \cdot \left(-\left(x + -1\right)\right) - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

      8. +-commutative 19.5%

         \[\leadsto \frac{1}{1 - x \cdot x} \cdot \left(-\color{blue}{\left(-1 + x\right)}\right) - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

      9. distribute-neg-in 19.5%

         \[\leadsto \frac{1}{1 - x \cdot x} \cdot \color{blue}{\left(\left(--1\right) + \left(-x\right)\right)} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

      10. metadata-eval 19.5%

          \[\leadsto \frac{1}{1 - x \cdot x} \cdot \left(\color{blue}{1} + \left(-x\right)\right) - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

      11. sub-neg 19.5%

          \[\leadsto \frac{1}{1 - x \cdot x} \cdot \color{blue}{\left(1 - x\right)} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

   5. Applied egg-rr 19.5%

      \[\leadsto \color{blue}{\frac{1}{1 - x \cdot x} \cdot \left(1 - x\right)} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

   6. Taylor expanded in x around inf 18.8%

      \[\leadsto \frac{1}{1 - x \cdot x} \cdot \left(1 - x\right) - \color{blue}{\frac{1}{x}} \]

   7. Taylor expanded in x around inf 62.7%

      \[\leadsto \color{blue}{\frac{-1}{{x}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 62.7%

         \[\leadsto \frac{-1}{\color{blue}{x \cdot x}} \]

   9. Simplified 62.7%

      \[\leadsto \color{blue}{\frac{-1}{x \cdot x}} \]

   if -1 < x < 1.14999999999999997e77

   1. Initial program 88.3%

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

   3. Simplified 88.3%

      \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]

   4. Taylor expanded in x around 0 86.7%

      \[\leadsto \color{blue}{1} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

   5. Taylor expanded in x around 0 86.3%

      \[\leadsto 1 - \color{blue}{\frac{2}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.15 \cdot 10^{+77}\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{x}\\ \end{array} \]

Alternative 5: 83.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ 1 + \left(-1 - \frac{2}{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 1.0 (- -1.0 (/ 2.0 x))))

C

double code(double x) {
	return 1.0 + (-1.0 - (2.0 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + ((-1.0d0) - (2.0d0 / x))
end function

Java

public static double code(double x) {
	return 1.0 + (-1.0 - (2.0 / x));
}

Python

def code(x):
	return 1.0 + (-1.0 - (2.0 / x))

Julia

function code(x)
	return Float64(1.0 + Float64(-1.0 - Float64(2.0 / x)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 + (-1.0 - (2.0 / x));
end

Wolfram

code[x_] := N[(1.0 + N[(-1.0 - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.6%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

3. Simplified 84.6%

   \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]

4. Taylor expanded in x around 0 54.3%

   \[\leadsto \color{blue}{1} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

5. Taylor expanded in x around 0 83.1%

   \[\leadsto 1 - \left(\frac{2}{x} - \color{blue}{-1}\right) \]

6. Final simplification 83.1%

   \[\leadsto 1 + \left(-1 - \frac{2}{x}\right) \]

Alternative 6: 52.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{-2}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ -2.0 x))

C

double code(double x) {
	return -2.0 / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-2.0d0) / x
end function

Java

public static double code(double x) {
	return -2.0 / x;
}

Python

def code(x):
	return -2.0 / x

Julia

function code(x)
	return Float64(-2.0 / x)
end

MATLAB

function tmp = code(x)
	tmp = -2.0 / x;
end

Wolfram

code[x_] := N[(-2.0 / x), $MachinePrecision]

Derivation

1. Initial program 84.6%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

3. Simplified 84.6%

   \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]

4. Taylor expanded in x around 0 54.9%

   \[\leadsto \color{blue}{\frac{-2}{x}} \]

5. Final simplification 54.9%

   \[\leadsto \frac{-2}{x} \]

Alternative 7: 3.3% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -1.0)

C

double code(double x) {
	return -1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -1.0d0
end function

Java

public static double code(double x) {
	return -1.0;
}

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

function tmp = code(x)
	tmp = -1.0;
end

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 84.6%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

3. Simplified 84.6%

   \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]

4. Taylor expanded in x around 0 54.2%

   \[\leadsto \frac{1}{1 + x} - \left(\frac{2}{x} - \color{blue}{-1}\right) \]

5. Taylor expanded in x around inf 3.3%

   \[\leadsto \color{blue}{-1} \]

6. Final simplification 3.3%

   \[\leadsto -1 \]

Developer target: 99.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{x \cdot \left(x \cdot x - 1\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (* x (- (* x x) 1.0))))

C

double code(double x) {
	return 2.0 / (x * ((x * x) - 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (x * ((x * x) - 1.0d0))
end function

Java

public static double code(double x) {
	return 2.0 / (x * ((x * x) - 1.0));
}

Python

def code(x):
	return 2.0 / (x * ((x * x) - 1.0))

Julia

function code(x)
	return Float64(2.0 / Float64(x * Float64(Float64(x * x) - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (x * ((x * x) - 1.0));
end

Wolfram

code[x_] := N[(2.0 / N[(x * N[(N[(x * x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023275 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))